reserve X for set;
reserve X,X1,X2 for non empty set;
reserve S for SigmaField of X;
reserve S1 for SigmaField of X1;
reserve S2 for SigmaField of X2;
reserve M for sigma_Measure of S;
reserve M1 for sigma_Measure of S1;
reserve M2 for sigma_Measure of S2;

theorem
for E being Element of sigma measurable_rectangles(S1,S2), x be Element of X1
holds
  ( M2.Measurable-X-section(E,x) <> 0
     implies Integral2(M2,Xchi(E,[:X1,X2:])).x = +infty ) &
  ( M2.Measurable-X-section(E,x) = 0
     implies Integral2(M2,Xchi(E,[:X1,X2:])).x = 0 )
proof
    let E be Element of sigma measurable_rectangles(S1,S2),
    x be Element of X1;
    ProjPMap1(Xchi(E,[:X1,X2:]),x) = Xchi(X-section(E,x),X2)
      by MESFUN12:35; then
A1: Integral2(M2,Xchi(E,[:X1,X2:])).x
     = Integral(M2,Xchi(X-section(E,x),X2)) by MESFUN12:def 8;
A2: Measurable-X-section(E,x) = X-section(E,x) by MEASUR11:def 6;
    hence M2.Measurable-X-section(E,x) <> 0 implies
     Integral2(M2,Xchi(E,[:X1,X2:])).x = +infty by A1,MEASUR10:33;
    assume M2.Measurable-X-section(E,x) = 0;
    hence thesis by A1,A2,MEASUR10:33;
end;
